In this lecture we summarize some simple properties enjoyed by block matrices (also called partitioned matrices).
We are going to assume that the reader is already familiar with the concept of a block matrix.
If two block matrices and have the same dimension and are partitioned in the same way, we obtain their sum by adding the corresponding blocks.
Example If we can compute their sum as
All the couples of summands need to have the same dimension. For instance, in the example above, if is ( rows and columns), then must be .
This property of block matrices is a direct consequence of the definition of matrix addition. Two matrices having the same dimension can be added together by adding their corresponding entries. For example, the -th entry of is the sum of the -th entry of and the -th entry of . This does not change if we first partition and and then we add together the two blocks to which the -th entries of and respectively belong.
Remember that the multiplication of a matrix by a scalar is performed by multiplying all the entries of by the scalar .
The same result can be achieved by multiplying all the blocks of by (because then all the entries of each block are multiplied by )
Example If then
The multiplication of two block matrices can be carried out as if their blocks were scalars, by using the standard rule for matrix multiplication: the -th block of the product is equal to the dot product between the -th row of blocks of and the -th column of blocks of .
Example Given two block matriceswe have that
As all the products must be well-defined, all the couples of blocks involved in a multiplication must be conformable. For instance, in the example above the number of columns of and the number of rows of must coincide for the product to be well-defined.
A proof that the dot product formula can be applied to block matrices follows.
Let us start from the case of the two matrices and in the previous example. Suppose that the blocks and have columns. As a consequence, and must have rows for the block products to be well-defined. Further assume that the blocks and have columns. It follows that and must have rows. By the definition of matrix product, the -th entry of isNow, suppose that the partition of leaves rows in the upper part of the matrix and in the lower one. Further assume that the partition of leaves columns to the left and to the right. Then, if and (upper-left quadrant of ), we havewhere we have used the facts that: 1) when and ; when and ; 3) when and ; 4) when and . Thus, as far as the upper-left quadrant is concerned, the claim we wanted to prove is true. Along similar lines, we can discuss the case in which and (upper right quadrant), for whichWe do not report the proofs for the remaining two quadrants, which are analogous. Moreover, similar case-by-case discussions can be performed if the block matrix is partitioned in a different manner (i.e., it has different numbers of horizontal and vertical cuts).
The proof, although tedious, allows us to better understand under what condition all the blocks can be multiplied. The partitions need to be such that a vertical partition of leaves columns to the left and to the right if and only if an horizontal partition of leaves rows in the upper part of the matrix and in the lower part. There are no constraints on the horizontal partitions of and the vertical partitions of .
The transpose of a block-matrix is the matrix such that the -th block of is equal to the transpose of the -th block of .
Example The transpose of the partitioned matrix is
A proof follows.
Let us first prove the result for the matrix in the example. Suppose that: 1) and have rows; 2) and have rows; 3) and have columns; 4) and have columns. Then, we can define the -th entry of asBy the definition of transpose, we have that the -th entry of is Therefore,We can easily check that this case-by-case definition corresponds to The proofs for block matrices having different partitions (i.e., different numbers of horizontal and vertical cuts) are similar (the case-by-case definitions of the matrices change based on the number of cuts).
Below you can find some exercises with explained solutions.
Define the block matrixwhere is an identity matrix and is a matrix of zeros. Compute the product .
We have that the transpose of isand the product is
Define the block matrixHow is the transpose structured?
The transpose of is
Please cite as:
Taboga, Marco (2017). "Properties of block matrices", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/properties-of-block-matrices.
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